We have developed boundary controllers for a cantilevered Euler-Bernoulli Beam with point mass dynamics at the free-end. Specifically, we have developed an exact model knowledge controller which exponentially stabilizes the beam position and an adaptive controller which asymptotically stabilizes the beam position. Furthermore, to validate the proposed control schemes, we have designed and built an experimental test stand. The experimental test stand consists of the following components:
The controller was implemented on a Windows-based GUI. The data acquisition was done using a C30 DSP board and a DS2 motion control board. The control program was written in C and a software-based commutation scheme was used to apply the desired force to the end of the beam.
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Many mechanical systems exhibit elastic effects which are often modeled using a linear partial differential equations(PDE) and a set of boundary conditions. Since there has been little control synthesis work for PDE based systems as compared to the abundance of control design techniques available for ordinary differential equations (ODE), some of the previous approaches for elastic systems rely on discretizing the PDE model into a set of ODEs. Unfortunately, a stability result generated for a discreatized ODE model under a proposed control cannot be generalized to the PDE model under the same control. That is, the neglected, higher order modes could possibly destabilize the mechanical system under a discretized model based control. Since the actual number of modes in an elastic system is infinite, it is often not clear how many modes should be used during the construction of the discretized ODE model. In addition, if a large number of modes are utilized to approximate a PDE-based system, the order of the discretized linear ODE model is often relatively high; hence, the resulting controller can be a complex high order algorithm. In contrast, boundary controllers designed for the non-discretized PDE model are often simple compensators which insure closed-loop stability for an infinite number of modes.
In the proposed work, we develop input force control strategies at the free-end of a cantilevered Euler-Bernoulli beam with point-mass dynamics at the free-end. Specifically, we develop an exact model knowledge controller which exponentially stabilizes the position of the beam given exact knowledge of some of the mechanical system parameters and measurements of the beam's free-end sheer, sheer-rate, and velocity. We also illustrate how the exact model knowledge controller can be redesigned as an adaptive controller which asymptotically stabilizes the position of the beam while compensating for parametric uncertainty. The control approach differs from the previous work in that: i) the controller compensates for point-mass dynamics on the boundary, ii) the stability analysis utilizes relatively simple mathematical tools to illustrate the exponential and asymptotic stability results, and iii) this work seems to be the first to fuse adaptive nonlinear ODE techniques with PDE boundary control techniques.
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